VIBRATIONS OF ELASTIC PLATES AND SHELLS OF VARIABLE THICKNESS

Abstract

This thesis is an attempt to study some of the problems on the vibration of elastic plates and shells of variable thickness. It consists of eight chapters. The first seven chapters deal with the free transverse vibrations of rectan gular and circular plates of linearly or parabolically varying thickness. The last chapter deals with the free torsional vibrations of conical and cylindrical shells of variable thickness. In all the eases the frequency equation and the defleetion function for the various modes of vibration have been derived. lumerical values of the frequencies and the mode shapes have been obtained in most of the oases for the first three modes of vibration. Chapter-wise summary of the thesis is given below. Chanter I free transverse vibrations of an elastic plate of infinite length and finite width with parabolically varying thickness have been studied on the basis of olassioal theory of plates. The solution of the equation of motion is obtained by Probcnius method. The deflection of the plate is obtained as a power series. Numerical results for the frequency parameter have been computed for different combinations of boundary conditions and for different values of taper constant. Chapter II Effect of transverse shear and rotatory inertia on vibrations of infinite plates of parabolically varying thickness has been studied. The differential equations governing the 11 motion of rectangular plates derived by Mlndlin have been extended for plates of variable thickness and solved by irobenius method. Transverse deflection as well as rotation in the plate elements are obtained as Infinite series. 1requency parameters are calculated for different values of taper constsnt snd thickness to breadth ratio for two combi nations of boundary conditions. Chapter III Pree transverse vibrations of rectangular plates of parabolically varying thickness have been studied on the basis of classical theory of plates. One pair of parallel edges of the plate are taken to be simply supported while the remaining two edges are variously supported (clamped, free or simply supported). The solution of the equation of motion is obtained by Frobenius method. The deflection of the plate is obtained as the product of an infinite series and a function satisfying boundary conditions at the pair of simply supported edges. Numerical results for the frequency parameter have been computed for different combinations of boundary conditions for various values of taper constant and length to breadth ratio. Chanter IV The lowest natural frequency of square plates of parabolically varying thickness with all the four corners point supported has been studied. The approximate frequency equation is determined by using the Rayleigh technique. Numerical values for the frequency parameter have been computed for different values of the taper constant. Vl'l Chapter V free axisymmetric vibrations of circular plates of linearly varying thickness have been studied on the basis ef classical theory of plates. The differential equation governing the motion of such plates has been solved by Irobenius method. The transverse displacement has been expressed as an infinite series in the radial coordinate. Frequency parameters have been computed for the first three modes of vibration for clamped as well as simply supported circular plates for different values of taper constant* Chapter VI Effect of transverse shear and rotatory Inertia on free axisymmetric transverse vibrations of circular 'plates of linearly varying thickness has been studied. The Mindlin's equations of motion have been extended for plates of variable thickness, and solved by irobenius method. The transverse deflection as well as the rotation of the plate element are expressed as an infinite series in the radial coordinate. Numerical results for frequency parameter are computed for clamped as well as simply supported plates for different values of taper constant and thickness to radius ratio. Comparison has been made with the results obtained in Chapter V. Chapter VII Effect of transverse shear and rotatory inertia on free axisymmetric vibrations of circular plates of parabolically varying thickness has been studied with the help of the \V equations of motion obtained in chapter VI. The solution of the equations of motion has been obtained by Frobenius method. The transverse deflection as well as the rotation of the plate element are obtained as an infinite series in the radial coordinate. Frequency parameters are computed for clamped as well as simply supported plates for different values of taper constant and thickness to radius ratio. Comparison has been made with corresponding results for classical theory of plates. Chapter VIII Free torsional vibrations of conical and cylindrical shells of thickness varying as a power of distance have been discussed. The solutions depend upon a linear differential equation of second order which can be reduced to Bessel's equation. The governing equation is solved for both edges fixed. The numerical values of the frequency parameter for the first three modes of vibration are computed for shells of linearly and parabolically varying thickness for several different ratios of terminal radii. The extensive numerical work involved in this thesis was done on IBM 1620 Computer at the Computer Centre, Structural Engineering Research Centre, Roorkee and IBM 360 Computer at the Computer Centre, Delhi school of Economics, Delhi University, Delhi.